R help - Sep 2017 - Interesting behavior of lm() with small, problematic data sets

I've recently come across the following results reported from the lm()
function when applied to a particular type of admittedly difficult data.  When
working with
small data sets (for instance 3 points) with the same response for different
predicting variable, the resulting slope estimate is a reasonable approximation
of the expected 0.0, but the p-value of that slope estimate is a surprising
value.  A reproducible example is included below, along with the output of the
summary of results

######### example code
x <- c(1,2,3)
y <- c(1,1,1)

#above results in{ (1,1) (2,1) (3,1)} data set to regress

new.rez <- lm (y ~ x) # regress constant y on changing x)
summary(new.rez) # display results of regression

######## end of example code

Results:

Call:
lm(formula = y ~ x)

Residuals:
         1          2          3
 5.906e-17 -1.181e-16  5.906e-17

Coefficients:
              Estimate Std. Error    t value Pr(>|t|)
(Intercept)  1.000e+00  2.210e-16  4.525e+15   <2e-16 ***
x           -1.772e-16  1.023e-16 -1.732e+00    0.333
---
Signif. codes:  0 ?***? 0.001 ?**? 0.01 ?*? 0.05 ?.? 0.1 ? ? 1

Residual standard error: 1.447e-16 on 1 degrees of freedom
Multiple R-squared:  0.7794,    Adjusted R-squared:  0.5589
F-statistic: 3.534 on 1 and 1 DF,  p-value: 0.3112

Warning message:
In summary.lm(new.rez) : essentially perfect fit: summary may be unreliable


##############

There is a warning that the summary may be unreliable sue to the essentially
perfect fit, but a p-value of 0.3112 doesn?t seem reasonable.
As a side note, the various r^2 values seem odd too.







Tim Glover
Senior Scientist II (Geochemistry, Statistics), Americas - Environment &
Infrastructure, Amec Foster Wheeler
271 Mill Road, Chelmsford, Massachusetts, USA 01824-4105
T +01 978 692 9090      D +01 978 392 5383      M +01 850 445 5039
tim.glover at amecfw.com      amecfw.com


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Why does an unreliable fit have to provide "reasonable" results?

More specifically, p-values arise from observed distributions... if your slopes
are "in the noise" then the slope estimate's location within that
distribution could be anywhere relative to the center and spread of that very
narrow distribution, leading to, ah, what was it... oh, right...
"unreliable" results.
-- 
Sent from my phone. Please excuse my brevity.

On September 5, 2017 6:24:30 AM PDT, "Glover, Tim" <Tim.Glover at
amecfw.com> wrote:
>I've recently come across the following results reported from the lm()
>function when applied to a particular type of admittedly difficult
>data.  When working with
>small data sets (for instance 3 points) with the same response for
>different predicting variable, the resulting slope estimate is a
>reasonable approximation of the expected 0.0, but the p-value of that
>slope estimate is a surprising value.  A reproducible example is
>included below, along with the output of the summary of results
>
>######### example code
>x <- c(1,2,3)
>y <- c(1,1,1)
>
>#above results in{ (1,1) (2,1) (3,1)} data set to regress
>
>new.rez <- lm (y ~ x) # regress constant y on changing x)
>summary(new.rez) # display results of regression
>
>######## end of example code
>
>Results:
>
>Call:
>lm(formula = y ~ x)
>
>Residuals:
>         1          2          3
> 5.906e-17 -1.181e-16  5.906e-17
>
>Coefficients:
>              Estimate Std. Error    t value Pr(>|t|)
>(Intercept)  1.000e+00  2.210e-16  4.525e+15   <2e-16 ***
>x           -1.772e-16  1.023e-16 -1.732e+00    0.333
>---
>Signif. codes:  0 ?***? 0.001 ?**? 0.01 ?*? 0.05 ?.? 0.1 ? ? 1
>
>Residual standard error: 1.447e-16 on 1 degrees of freedom
>Multiple R-squared:  0.7794,    Adjusted R-squared:  0.5589
>F-statistic: 3.534 on 1 and 1 DF,  p-value: 0.3112
>
>Warning message:
>In summary.lm(new.rez) : essentially perfect fit: summary may be
>unreliable
>
>
>##############
>
>There is a warning that the summary may be unreliable sue to the
>essentially perfect fit, but a p-value of 0.3112 doesn?t seem
>reasonable.
>As a side note, the various r^2 values seem odd too.
>
>
>
>
>
>
>
>Tim Glover
>Senior Scientist II (Geochemistry, Statistics), Americas - Environment
>& Infrastructure, Amec Foster Wheeler
>271 Mill Road, Chelmsford, Massachusetts, USA 01824-4105
>T +01 978 692 9090      D +01 978 392 5383      M +01 850 445 5039
>tim.glover at amecfw.com      amecfw.com
>
>
>This message is the property of Amec Foster Wheeler plc and/or its
>subsidiaries and/or affiliates and is intended only for the named
>recipient(s). Its contents (including any attachments) may be
>confidential, legally privileged or otherwise protected from disclosure
>by law. Unauthorised use, copying, distribution or disclosure of any of
>it may be unlawful and is strictly prohibited. We assume no
>responsibility to persons other than the intended named recipient(s)
>and do not accept liability for any errors or omissions which are a
>result of email transmission. If you have received this message in
>error, please notify us immediately by reply email to the sender and
>confirm that the original message and any attachments and copies have
>been destroyed and deleted from your system. If you do not wish to
>receive future unsolicited commercial electronic messages from us,
>please forward this email to: unsubscribe at amecfw.com and include
>?Unsubscribe? in the subject line. If applicable, you will continue to
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>
>______________________________________________
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>and provide commented, minimal, self-contained, reproducible code.

> On Sep 5, 2017, at 6:24 AM, Glover, Tim <Tim.Glover at amecfw.com>
wrote:
> 
> I've recently come across the following results reported from the lm()
function when applied to a particular type of admittedly difficult data.  When
working with
> small data sets (for instance 3 points) with the same response for
different predicting variable, the resulting slope estimate is a reasonable
approximation of the expected 0.0, but the p-value of that slope estimate is a
surprising value.  A reproducible example is included below, along with the
output of the summary of results
> 
> ######### example code
> x <- c(1,2,3)
> y <- c(1,1,1)
> 
> #above results in{ (1,1) (2,1) (3,1)} data set to regress
> 
> new.rez <- lm (y ~ x) # regress constant y on changing x)
> summary(new.rez) # display results of regression
> 
> ######## end of example code
> 
> Results:
> 
> Call:
> lm(formula = y ~ x)
> 
> Residuals:
>         1          2          3
> 5.906e-17 -1.181e-16  5.906e-17
> 
> Coefficients:
>              Estimate Std. Error    t value Pr(>|t|)
> (Intercept)  1.000e+00  2.210e-16  4.525e+15   <2e-16 ***
> x           -1.772e-16  1.023e-16 -1.732e+00    0.333
> ---
> Signif. codes:  0 ?***? 0.001 ?**? 0.01 ?*? 0.05 ?.? 0.1 ? ? 1
> 
> Residual standard error: 1.447e-16 on 1 degrees of freedom
> Multiple R-squared:  0.7794,    Adjusted R-squared:  0.5589
> F-statistic: 3.534 on 1 and 1 DF,  p-value: 0.3112
> 
> Warning message:
> In summary.lm(new.rez) : essentially perfect fit: summary may be unreliable
> 
> 
> ##############
> 
> There is a warning that the summary may be unreliable sue to the
essentially perfect fit, but a p-value of 0.3112 doesn?t seem reasonable.
> As a side note, the various r^2 values seem odd too.
You have an overfitted model with only 3 perfectly fit-able data points and you
are whinging about a Wald statistic about which you were warned. I think you are
wasting our time. (But I'm fully retired and I have a lot of time to waste.)

I seem to remember that a t-distribution with 1 degree of freedom is actually
the Cauchy distribution. I would point out that you can also get:
> 2*pt(-1.732e+00, 1)
[1] 0.3333414

So maybe from that perspective any value might be "reasonable" from
the perspective that you have that particular number data points (so one degree
of freedom) and are using an estimate of the t-statistic which is essentially
the ratio of 0/0 from a numerical point of view.

-- 
David.

Tim,

I think what you're seeing is 
https://en.wikipedia.org/wiki/Loss_of_significance.

Cheers,

Mark



From:   "Glover, Tim" <Tim.Glover at amecfw.com>
To:     "r-help at r-project.org" <r-help at r-project.org>
Date:   09/05/2017 11:37 AM
Subject:        [R] Interesting behavior of lm() with small, problematic 
data sets
Sent by:        "R-help" <r-help-bounces at r-project.org>



I've recently come across the following results reported from the lm() 
function when applied to a particular type of admittedly difficult data. 
When working with
small data sets (for instance 3 points) with the same response for 
different predicting variable, the resulting slope estimate is a 
reasonable approximation of the expected 0.0, but the p-value of that 
slope estimate is a surprising value.  A reproducible example is included 
below, along with the output of the summary of results

######### example code
x <- c(1,2,3)
y <- c(1,1,1)

#above results in{ (1,1) (2,1) (3,1)} data set to regress

new.rez <- lm (y ~ x) # regress constant y on changing x)
summary(new.rez) # display results of regression

######## end of example code

Results:

Call:
lm(formula = y ~ x)

Residuals:
         1          2          3
 5.906e-17 -1.181e-16  5.906e-17

Coefficients:
              Estimate Std. Error    t value Pr(>|t|)
(Intercept)  1.000e+00  2.210e-16  4.525e+15   <2e-16 ***
x           -1.772e-16  1.023e-16 -1.732e+00    0.333
---
Signif. codes:  0 ?***? 0.001 ?**? 0.01 ?*? 0.05 ?.? 0.1 ? ? 1

Residual standard error: 1.447e-16 on 1 degrees of freedom
Multiple R-squared:  0.7794,    Adjusted R-squared:  0.5589
F-statistic: 3.534 on 1 and 1 DF,  p-value: 0.3112

Warning message:
In summary.lm(new.rez) : essentially perfect fit: summary may be 
unreliable


##############

There is a warning that the summary may be unreliable sue to the 
essentially perfect fit, but a p-value of 0.3112 doesn?t seem reasonable.
As a side note, the various r^2 values seem odd too.







Tim Glover
Senior Scientist II (Geochemistry, Statistics), Americas - Environment & 
Infrastructure, Amec Foster Wheeler
271 Mill Road, Chelmsford, Massachusetts, USA 01824-4105
T +01 978 692 9090      D +01 978 392 5383      M +01 850 445 5039
tim.glover at amecfw.com      amecfw.com


This message is the property of Amec Foster Wheeler plc and/or its 
subsidiaries and/or affiliates and is intended only for the named 
recipient(s). Its contents (including any attachments) may be 
confidential, legally privileged or otherwise protected from disclosure by 
law. Unauthorised use, copying, distribution or disclosure of any of it 
may be unlawful and is strictly prohibited. We assume no responsibility to 
persons other than the intended named recipient(s) and do not accept 
liability for any errors or omissions which are a result of email 
transmission. If you have received this message in error, please notify us 
immediately by reply email to the sender and confirm that the original 
message and any attachments and copies have been destroyed and deleted 
from your system. If you do not wish to receive future unsolicited 
commercial electronic messages from us, please forward this email to: 
unsubscribe at amecfw.com and include ?Unsubscribe? in the subject line. If 
applicable, you will continue to receive invoices, project communications 
and similar factual, non-commercial electronic communications.

Please click http://amecfw.com/email-disclaimer for notices and company 
information in relation to emails originating in the UK, Italy or France.

______________________________________________
R-help at r-project.org mailing list -- To UNSUBSCRIBE and more, see
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide 
http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.


	[[alternative HTML version deleted]]

> I think what you're seeing is
> https://en.wikipedia.org/wiki/Loss_of_significance.
Almost. 
All the results in the OP's summary are reflections of finite precision in
the analytically exact solution, leading to residuals smaller than the double
precision limit. The summary is correctly warning that it's all potentially
nonsense, and indeed the only things you can trust are the coefficient values
(to within .Machine$double.eps or thereabouts)

Interestingly, though, my current version of R (3.4.0) gives numerically exact
coefficients (c(1,0) and identically zero standard errors.

So this particular example is apparently version-specific.

S Ellison


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